Recall that a singleton Grothendieck pretopology (henceforth 'singleton pretopology') on a category $C$ is a collection of maps $J$ containing the isomorphisms, closed under composition and stable under pullback (i.e. pullbacks of them exist, and they are stable). Each map is to be considered a covering family with a single element.

Consider the following two conditions:

$J$ is saturated: If $U \to V\to X$ is in $J$ then $V \to X$ is in $J$.

$J$ is admissible: $J$ contains the split epimorphisms, and if $U \to V\to X$ is in $J$ and $U \to V$ is a split epimorphism, then $V\to X$ is in $J$.

Now clearly saturated singleton pretopologies are admissible (notice saturated implies $J$ contains the split epis). An example of a saturated pretopology is the class ($K$-epi) of $K$-epimorphisms in a fintely complete category: maps $p:Q\to X$ such that there is a $K$-cover $k:U\to X$ and a map $s:U\to Q$ with $p\circ s = k$ (or more generally local sections for any pretopology, not necc. singleton). Now my question is this:

What is an example of an admissible singleton pretopology which is not of the form ($K$-epi) for some other pretopology $K$?

I know the general situation in categories like $Top$, $Diff$, but not in algebraic settings. As far as I know, pretopologies like ($K$-epi) don't turn up much (I may be wrong, and happy to be corrected), but do admissible singleton pretopologies otherwise arise in algebraic geometry?

I've found in the past here that people basically never answer questions about grothendieck topologies. In fact, I asked a question recently concerning them, and I got a bunch of votes up after I removed the topos-theory tag. The trick is to make your question look like it's about algebraic geometry. Also, the definition you're using of a singleton topology is confusing (you should note that the morphisms are supposed to be single-morphism covering-families).
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Harry GindiOct 17 '10 at 12:36

But I gave you a +1 anyway because I don't want to sound like a complainer =p.
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Harry GindiOct 17 '10 at 12:37

3

I think other people actually filter MO by tags, as opposed to scan all the questions for interesting things like some of us.
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David RobertsOct 17 '10 at 21:39